## Kirchhoff 型方程正规化解的多重性及渐近行为

1山西师范大学数学与计算机科学学院 太原 030031

2兰州大学数学与统计学院 兰州 730000

3华东师范大学数学科学学院, 数学与工程应用教育部重点实验室 &上海市核心数学与实践重点实验室 上海 200241

## Multiplicity and Asymptotic Behavior of Normalized Solutions for Kirchhoff-Type Equation

Jin Zhenfeng1,2, Sun Hongrui,2,*, Zhang Weimin3

1School of Mathematics and Computer Science, Shanxi Normal University, Taiyuan 030031

2School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000

3School of Mathematical Sciences, Key Laboratory of Mathematics and Engineering Applications (Ministry of Education) & Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241

 基金资助: 山西省基础研究计划项目(202303021212160)国家自然科学基金(11671181)甘肃省科技计划项目 (基础研究创新群体)(21JR7RA535)

 Fund supported: NSF of Shanxi Province(202303021212160)NSFC(11671181)Science Technology Program of Gansu Province(21JR7RA535)

$\begin{cases} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx\right)\Delta u=\lambda u+|u|^{p-2}u, \quad x\in \mathbb{R}^{3},\\ \|u\|^2_{2}=\rho,\end{cases}$

Abstract

In this paper, we consider the following Kirchhoff-type equation

$\begin{cases} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\ d x\right)\Delta u=\lambda u+|u|^{p-2}u \quad \mathrm{in}\ \mathbb{R}^{3},\\ \|u\|^2_{2}=\rho,\end{cases}$

where $a$, $b$, $\rho>0$ and $\lambda\in\mathbb{R}$ arises as Lagrange multiplier with respect to the mass constraint $\|u\|^2_{2}=\rho$. When $p\in\left(2,\frac{10}{3}\right)$ or $p\in\left(\frac{14}{3},6\right)$, we establish the existence of infinitely many radial $L^2$-normalized solutions by using the genus theory. Furthermore, we testify an asymptotic behavior of the above solutions with respect to the parameter $b\rightarrow 0^+$.

Keywords： Kirchhoff equation; Variational method; Normalized solution; Asymptotic behavior

Jin Zhenfeng, Sun Hongrui, Zhang Weimin. Multiplicity and Asymptotic Behavior of Normalized Solutions for Kirchhoff-Type Equation[J]. Acta Mathematica Scientia, 2024, 44(4): 871-884

## 1 引言

$-\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2} dx\right)\Delta u=\lambda u+f(u), \quad x\in\mathbb{R}^{3},$

$\|u\|^2_{2}=\rho$

$\begin{cases} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\ d x \right)\Delta u=\lambda u+|u|^{p-2}u, \quad x\in\mathbb{R}^{3}, \\\|u\|^2_{2}=\rho, \rho>0,\end{cases}$

\begin{aligned}I(u)=\frac{a}{2}\int_{\mathbb{R}^{3}}|\nabla u|^{2} \ d x+\frac{b}{4}\left(\int_{\mathbb{R}^{3}}|\nabla u|^{2} \ d x\right)^{2}-\frac{1}{p}\int_{\mathbb{R}^{3}}|u|^p \ d x, u\in H^{1}(\mathbb{R}^{3})\end{aligned}

$-a \Delta u=\lambda u+|u|^{p-2} u, \quad x\in\mathbb{R}^{3}.$

$\bullet$$H^{1}(\mathbb{R}^{3}):=\left\{u\in L^{2}(\mathbb{R}^{3}):\nabla u\in L^{2}(\mathbb{R}^{3})\right\} 是通常的 Sobolev 空间并赋予范数 \|u\|:=\left(\|\nabla u\|^2_{2}+\|u\|^2_{2}\right)^{\frac{1}{2}}; \bullet$$H^{1}_{r}(\mathbb{R}^{3}):=\left\{u \in H^{1}(\mathbb{R}^{3}): u \text { 是径向对称的}\right\}$;

$\bullet$$\rightharpoonup 表示相应的函数空间上的弱收敛. ## 2 预备知识 在本节中, 我们给出一些预备引理. 首先有如下 Gagliardo-Nirenberg 不等式[38]. 引理 2.12\leq q< 2^{*}, 则对任意的 u\in H^{1}(\mathbb{R}^3), 存在仅依赖于 q 的常数 C_{q}, 使得 \|u\|_{q}\leq C_{q}\|\nabla u\|^{\gamma_{q}}_{2} \|u\|^{1-\gamma_{q}}_{2}, 其中 \gamma_{q}=\frac{3(q-2)}{2q}. 对任意的 s\in\mathbb{R}$$u\in H^{1}(\mathbb{R}^{3})$, 定义

$s\star u(x):={\rm e}^{\frac{3}{2}s}u({\rm e}^{s}x).$

$s\star u\in H^{1}(\mathbb{R}^{3}), \|s\star u\|_{2}=\|u\|_{2}.$

(i) 存在唯一的实数 $s(u)\in\mathbb{R}$ 使得 $P(s(u)\star u)=0$.

(ii) 若 $p\in\left(2,\frac{10}{3}\right)$, 则对任意的 $s\neq s(u)$, 有 $I(s(u)\star u)<I(s\star u)$, 且 $I(s(u)\star u)<0$.$p\in\left(\frac{14}{3},6\right)$, 则对任意的 $s\neq s(u)$, 有 $I(s(u)\star u)>I(s\star u)$, 且 $I(s(u)\star u)>0$.

(iii) 映射 $u\mapsto s(u)$ 关于 $u\in H^{1}(\mathbb{R}^{3})\setminus\{0\}$ 是连续的.

(iv) 对任意的 $y\in\mathbb{R}^{3}$, 有 $s(u(\cdot+y))=s(u(\cdot))$. 进一步, $s(-u)=s(u)$.

(i) 由于

$I(s\star u)=\frac{a}{2}{\rm e}^{2s} \|\nabla u\|^{2}_{2}+\frac{b}{4}{\rm e}^{4s} \|\nabla u\|^{4}_{2}-\frac{{\rm e}^{\gamma_{p}ps}}{p} \|u\|^{p}_{p},$

$p\in\left(\frac{14}{3},6\right)$ 时, 由 (2.1) 式, 可得 I\mathcal{P}_{\rho} 上是强制的. 对于给定的 \rho>0, 定义辅助泛函 J:S_{\rho}\rightarrow \mathbb{R} 如下 \begin{aligned}J(u)=I(s(u)\star u)=\frac{a}{2}{\rm e}^{2s(u)} \|\nabla u\|^{2}_{2}+\frac{b}{4}{\rm e}^{4s(u)} \|\nabla u\|^{4}_{2}-\frac{{\rm e}^{p\gamma_{p}s(u)}}{p} \|u\|^{p}_{p},\end{aligned} 其中 s(u) 是由引理 2.2(i) 给出的. 通过与[19,引理 4.2,引理 4.3] 相似的讨论, 可得如下结果. 引理 2.4 假设 a, b>0, p\in\left(2,\frac{10}{3}\right) 或者 p\in\left(\frac{14}{3},6\right). 则泛函 J: S_{\rho} \rightarrow \mathbb{R}\mathcal{C}^{1} 的, 且对任意的 $u\in S_{\rho}$$\varphi\in T_{u}S_{\rho}, 有 {\rm d}J(u)[\varphi]={\rm d}I(s(u)\star u)[s(u)\star\varphi], 其中 T_{u} S_{\rho}:=\left\{z\in H^{1}(\mathbb{R}^{3}):\int_{\mathbb{R}^{3}}zu=0\right\}. ## 3 主要结果的证明 在本节中, 首先研究问题 (1.4) 的 L^2-正规化解的多重性. 利用极小极大原理 (参见文献[12,定理 7.2]), 我们将构造泛函 J 在能量水平 m_{\rho,k} 上的一个 Palais-Smale 序列 \{u^{k}_{n}\}^{\infty}_{n=1}\subset\mathcal{P}_{\rho,r}, 即满足本节中的 (3.2) 式. X\subset H^{1}(\mathbb{R}^{3}), \sigma: H^{1}(\mathbb{R}^{3})\rightarrow H^{1}(\mathbb{R}^{3}) 且满足 \sigma(u)=-u.\sigma(A)=A, 则称集合 A\subset X$$\sigma$-不变的. 对任意的 $(t,u)\in[0,1]\times X$, 若 $\eta(t,\sigma(u))=\sigma(\eta(t,u))$, 则称同伦 $\eta:[0,1]\times X\rightarrow X$$\sigma-等变的. 文献[12,定义 7.1] 中给出如下定义. 定义 3.1 假设 B$$X\subset H^{1}(\mathbb{R}^{3})$ 的一个闭的 $\sigma$-不变集. $X$ 的一类紧子集族 $\mathcal{G}$ 称为带闭的边界 $B$$\sigma-同伦稳定族, 若 \mathcal{G} 满足 (i) \mathcal{G} 中的任一集合是 \sigma-不变的; (ii) \mathcal{G} 中的任一集合包含 B; (iii) 假设 A$$\mathcal{G}$ 中的任一集合, $\eta\in C([0,1]\times X,X)$$\sigma-同伦等变的, 且对任意的 (t,u)\in(\{0\}\times X)\cup([0,1]\times B), 满足 \eta(t,u)=u. 则有 \eta(\{1\}\times A)\in\mathcal{G}. 引理 3.1 假设 a, b>0, p\in\big(2,\frac{10}{3}\big).\mathcal{G}$$S_{\rho,r}$ 的带闭的边界 B=\emptyset\sigma-同伦稳定紧子集族, 且令 m_{\rho,\mathcal{G}}:=\inf_{A\in\mathcal{G}}\max_{u\in A} J(u). m_{\rho,\mathcal{G}}<0, 则泛函 I 在能量水平 m_{\rho,\mathcal{G}} 上存在一个 Palais-Smale 序列 \{u_{n}\}\subset\mathcal{P}_{\rho,r}. 显然存在序列 \{A_n\}\subset\mathcal{G} 使得 \max_{u\in A_n}J(u)\leq m_{\rho,\mathcal{G}}+\frac{1}{n}. 利用引理 2.2(iii), 定义连续映射 \eta 如下 \begin{aligned}\eta:[0,1]\times S_{\rho,r}&\rightarrow S_{\rho,r},\\\eta(t,u)&=(ts(u))\star u,\end{aligned} 且满足对任意的 (t,u)\in\{0\}\times S_{\rho,r}, 有 \eta(t,u)=u. 因此由 \mathcal{G} 的定义可知, 当 n\in\mathbb{N}^{+} 时, 有 \begin{aligned}D_{n}:=\eta(1,A_{n})=\{s(u)\star u: u\in A_{n}\}\in\mathcal{G}\end{aligned} D_{n}\subset\mathcal{P}_{\rho,r}.J 的定义可知, 对任意的 s\in\mathbb{R}u\in S_{\rho,r}, 有 $J(s\star u)=J(u)$. 因此,

\begin{aligned}\max_{D_{n}}J=\max_{A_{n}}J\rightarrow m_{\rho,\mathcal{G}}.\end{aligned}

$\{D_{n}\}\subset\mathcal{G}$ 同样是 $m_{\rho,\mathcal{G}}$ 的极小化序列. 接下来, 利用极小极大原理 (参见文献[12,定理 7.2]), 可得泛函 $J$ 在能量水平 $m_{\rho,\mathcal{G}}$ 上存在一个 Palais-Smale 序列 $\{v_{n}\}\subset S_{\rho,r}$, 且满足当 $n\rightarrow\infty$ 时, 有 $\mathrm{dist}_{H^{1}(\mathbb{R}^{3})}(v_{n},D_{n})\rightarrow 0$. 定义

$s_{n}:=s(v_{n}), u_{n}:=s_{n} \star v_{n}=s(v_{n}) \star v_{n}\in \mathcal{P}_{\rho,r}.$

$\|\nabla u_{n}\|_{2}\rightarrow0, J(v_n)=I(u_{n})=J(u_n)\rightarrow 0,$

$\liminf_{n\rightarrow\infty}\|\nabla u_{n}\|_{2} \geq 2\tau>0.$

\begin{aligned}{\rm e}^{-2s_{n}}=\frac{\int_{\mathbb{R}^{3}}|\nabla v_{n}|^{2} \ d x}{\int_{\mathbb{R}^{3}}|\nabla u_{n}|^{2} \ d x}.\end{aligned}

\begin{aligned}\max_{D_{n}}I=\max_{D_{n}} J \rightarrow m_{\rho,\mathcal{G}}.\end{aligned}

\begin{aligned} \mathrm{Ind}(A):=\min\left\{k\in \mathbb{N}^{+}: \exists\phi: A \rightarrow\mathbb{R}^{k}\backslash\{0\}, \phi \text {是连续的奇映射}\right\}. \end{aligned}

$\Sigma$$S_{\rho,r} 的一个 \sigma-不变紧子集族. 对任意的 k \in \mathbb{N}^{+}, 定义 \mathcal{G}_{k}:=\{A\in\Sigma\mid\mathrm{Ind}(A)\geq k\} m_{\rho,k}:=\inf_{A\in\mathcal{G}_{k}}\max_{u\in A} J(u). \{V_{k}\} 表示 H^{1}_{r}(\mathbb{R}^{3}) 的一列有限维线性子空间, 且满足 V_{k}\subset V_{k+1}, 维数 \dim V_{k}=k$$\mathop{\cup}\limits_{k\geq 1}V_{k}$$H^{1}_{r}(\mathbb{R}^{3}) 中稠密. 引理 3.2 假设 a, b>0, p\in\left(2,\frac{10}{3}\right). 则对任一 k\in\mathbb{N}^{+}, 有 (i) \mathcal{G}_{k}\neq\emptyset, \mathcal{G}_{k}$$S_{\rho,r}$ 的带闭的边界 $B=\emptyset$$\sigma-同伦稳定紧子集族. (ii) m_{\rho,k}\leq m_{\rho,k+1}<0. (i) 利用亏格的性质, 可得 \mathrm{Ind}(S_{\rho}\cap V_{k})=k. 因此 \mathcal{G}_{k} \neq \emptyset. 根据定义 3.1 以及亏格的性质, 可得 (i) 成立. (ii) 对任意的 u\in S_\rho, 利用引理 2.3(ii), 可知 J(u)=\underset{s\in\mathbb{R}}{\min} I(s\star u)<0. 假设存在 A\in \mathcal{G}_k, 使得 \underset{u\in A}{\max}J(u)=0. 由于 A$$S_{\rho,r}$ 中的紧集, 故存在 $u\in A$ 使得 $J(u)=0$, 这与 (3.3) 式矛盾. 因此, $m_{\rho,k}<0$. 根据 $\mathcal{G}_{k+1} \subset \mathcal{G}_{k}$, 可推出 $m_{\rho,k}\leq m_{\rho,k+1}$.

\begin{aligned} & u_n\rightharpoonup u \quad \text{于} H^{1}_{r}(\mathbb{R}^{3}),\\ & u_n\rightarrow u \quad \text{于} L^q(\mathbb{R}^{3}), q\in(2,6),\\ & u_n\rightarrow u \quad \mathrm{a.e.} \text{于} \mathbb{R}^3.\end{aligned}

\begin{aligned}&a\int_{\mathbb{R}^{3}} \nabla u_{n} \cdot \nabla \varphi \ d x+b\int_{\mathbb{R}^{3}} |\nabla u_{n}|^2 \ d x\int_{\mathbb{R}^{3}}\nabla u_{n}\cdot \nabla \varphi \ d x-\lambda_n\int_{\mathbb{R}^{3}} u_{n} \varphi \ d x\\&-\int_{\mathbb{R}^{3}}|u_{n}|^{p-2}u_{n}\cdot \varphi \ d x=o(1)\|\varphi\|,\quad\forall\varphi \in H^{1}(\mathbb{R}^{3}),\end{aligned}

$\lambda_n\rho =a\|\nabla u_{n}\|^{2}_{2}+b\|\nabla u_{n}\|^{4}_{2} -\|u_{n}\|^{p}_{p}+o(1).$

$(a+bB)\|\nabla(u_{n}-u)\|^{2}_{2}-\lambda\|u_{n}-u\|^{2}_{2}\rightarrow 0.$

$u_n\rightharpoonup u \text{于} H_r^1(\mathbb{R}^3),\quad u_n\rightharpoonup u \text{于} L^2(\mathbb{R}^3).$

$\left(\pi_{k_n} u_n, u\right)_{L^2(\mathbb{R}^3)}=\left(u_n, \pi_{k_n}u\right)_{L^2(\mathbb{R}^3)}\rightarrow (u, u)_{L^2(\mathbb{R}^3)}.$

$\pi_{k_n} u_n\rightarrow 0$$L^2(\mathbb{R}^3), 可得 \|u\|_{2}=0. 因此, u=0$$\|u_n\|_{p}\rightarrow 0$. 利用 $u_n\in \mathcal{P}_\rho$, 则有 $\|\nabla u_{n}\|_{2}\rightarrow 0$, $I(u_n)\rightarrow 0$, 这与 $I(u_n)\le c_0$ 矛盾. 证毕.

$\Psi(u)=\frac{\pi_{k(c_0)}u}{\left\|\pi_{k(c_0)}u\right\|}.$

$h_b(s)=a\|\nabla u\|^{2}_{2}+b{\rm e}^{2s} \|\nabla u\|^{4}_{2}-\gamma_{p}{\rm e}^{(\gamma_{p}p-2)s} \|u\|^{p}_{p}.$

$0<b_1<b_2$. 显然有 $h_{b_1}(s)\leq h_{b_2}(s)$, 因此, 当 $p\in\left(2,\frac{10}{3}\right)$ 时, 有 $s_{b_2}(u)\leq s_{b_1}(u)$; 当 $p\in\left(\frac{14}{3},6\right)$ 时, 有 $s_{b_1}(u)\leq s_{b_2}(u)$. 对任意的 $u\in A$, 可得

\begin{aligned}J_{b}(u)&=I_b(s_b(u)\star u)=\frac{a}{2}{\rm e}^{2s_b(u)} \|\nabla u\|^{2}_{2}+\frac{b}{4}{\rm e}^{4s_b(u)} \|\nabla u\|^{4}_{2}-\frac{{\rm e}^{\gamma_{p}ps_b(u)}}{p} \|u\|^{p}_{p}\\&=\frac{(3p-10)a}{6(p-2)}{\rm e}^{2s_b(u)} \|\nabla u\|^{2}_{2}+\frac{(3p-14)b}{12(p-2)}{\rm e}^{4s_b(u)} \|\nabla u\|^{4}_{2}.\end{aligned}

(i) 若 $p\in\left(2,\frac{10}{3}\right)$, 则对任一 $k\in \mathbb{N}^{+}$, 可推出

$-\infty<m^0_{\rho,k}\leq m^{b_{m}}_{\rho,k}=I_{b_{m}}\left(u^{b_{m}}_{k}\right)\leq I_{b_{m}}\left(u^{b_{m}}_{k+1}\right)=m^{b_{m}}_{\rho,k+1}\leq m^1_{\rho,k+1}<0;$

(ii) 若 $p\in\left(\frac{14}{3},6\right)$, 则对任一 $k\in \mathbb{N}^{+}$, 可推出

$0<m^0_{\rho,k}\leq m^{b_{m}}_{\rho,k}=I_{b_{m}}\left(u^{b_{m}}_{k}\right)\leq I_{b_{m}}\left(u^{b_{m}}_{k+1}\right)=m^{b_{m}}_{\rho,k+1}\leq m^1_{\rho,k+1}<+\infty.$

\begin{aligned}I_{b_{m}}\left(u^{b_{m}}_{k}\right)&=I_{b_{m}}\left(u^{b_{m}}_{k}\right)-\frac{2}{3(p-2)}P_{b_{m}}\left(u^{b_{m}}_{k}\right)\\&=\frac{(3p-10)a}{6(p-2)}\left\|\nabla u^{b_{m}}_{k}\right\|^{2}_{2}+\frac{(3p-14)b_{m}}{12(p-2)}\left\|\nabla u^{b_{m}}_{k}\right\|^{4}_{2},\end{aligned}

\begin{aligned}\lambda^{b_{m}}_{k}\rho=a\left\|\nabla u^{b_{m}}_{k}\right\|^{2}_{2}+b_{m}\left\|\nabla u^{b_{m}}_{k}\right\|^{4}_{2}-\left\|u^{b_{m}}_{k}\right\|^{p}_{p}.\end{aligned}

\begin{aligned}u^{b_{m}}_{k}&\rightharpoonup u^{0}_{k} \quad\text{ 于 }H^{1}_r(\mathbb{R}^{3}),\\u^{b_{m}}_{k}&\rightarrow u^{0}_{k} \quad\text{ 于 }L^q(\mathbb{R}^{3}), q\in(2,6),\\u^{b_{m}}_{k}&\rightarrow u^{0}_{k} \ \quad\mathrm{a.e.} \text{于} \mathbb{R}^3,\\\lambda^{b_{m}}_{k}&\rightarrow \lambda^{0}_{k}\quad\text{ 于 }\mathbb{R}.\end{aligned}

\begin{aligned}a\int_{\mathbb{R}^{3}} \nabla u^{0}_{k} \cdot \nabla \varphi \ d x-\lambda^{0}_{k}\int_{\mathbb{R}^{3}} u^{0}_{k} \varphi \ d x-\int_{\mathbb{R}^{3}}|u^{0}_{k}|^{p-2}u^{0}_{k}\cdot \varphi \ d x=0,\quad\forall\varphi \in H^{1}(\mathbb{R}^{3}).\end{aligned}

$-a \Delta u=\lambda u+|u|^{p-2} u, \quad x\in\mathbb{R}^{3}.$

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Alves C.

On existence of multiple normalized solutions to a class of elliptic problems in whole $\mathbb{R}^N$

Z Angew Math Phys, 2022, 73: Article 97

Alves C, Ji C, Miyagaki O.

Normalized solutions for a Schrödinger equation with critical growth in $\mathbb{R}^N$

Calc Var Partial Differential Equations, 2022, 61: Article 18

Bao W, Cai Y.

Mathematical theory and numerical methods for Bose-Einstein condensation

Kinet Relat Models, 2013, 6: 1-135

Bartsch T, de Valeriola S.

Normalized solutions of nonlinear Schrödinger equations

Arch Math, 2013, 100: 75-83

Bartsch T, Zhong X, Zou W.

Normalized solutions for a coupled Schrödinger system

Math Ann, 2021, 380: 1713-1740

Berestycki H, Lions P.

Nonlinear scalar field equations. II. Existence of infinitely many solutions

Arch Rational Mech Anal, 1983, 82: 347-375

Cai L, Zhang F.

Normalized solutions of mass supercritical Kirchhoff equation with potential

J Geom Anal, 2023, 33: Article 107

Cao X, Xu J, Wang J.

The existence of solutions with prescribed $L^2$-norm for Kirchhoff type system

J Math Phys, 2017, 58: 041502

Carrião P, Miyagaki O, Vicente A.

Normalized solutions of Kirchhoff equations with critical and subcritical nonlinearities: the defocusing case

Partial Differ Equ Appl, 2022, 3: Article 64

Chen W, Huang X.

The existence of normalized solutions for a fractional Kirchhoff-type equation with doubly critical exponents

Z Angew Math Phys, 2022, 73: Article 226

Du M, Li F, Wang Z.

Multiplicity of normalized solutions for nonlinear Schrödinger-Poisson equation with Hardy potential

Acta Math Sci, 2022, 42A(2): 442-453

Ghoussoub N. Duality and Perturbation Methods in Critical Point Theory. Cambridge: Cambridge University Press, 1993

He Q, Lv Z, Zhang Y, Zhong X.

Existence and blow up behavior of positive normalized solution to the Kirchhoff equation with general nonlinearities: Mass super-critical case

J Differential Equations, 2023, 356: 375-406

Hu J, Mao A.

Normalized solutions to the Kirchhoff equation with a perturbation term

Differential Integral Equations, 2023, 36: 289-312

Hu T, Tang C.

Limiting behavior and local uniqueness of normalized solutions for mass critical Kirchhoff equations

Calc Var Partial Differential Equations, 2021, 60: Article 210

Huang X, Zhang Y.

Existence and uniqueness of minimizers for $L^2$-constrained problems related to fractional Kirchhoff equation

Math Methods Appl Sci, 2020, 43: 8763-8775

Jeanjean L.

Existence of solutions with prescribed norm for semilinear elliptic equations

Nonlinear Anal, 1997, 28: 1633-1659

Jeanjean L, Le T.

Multiple normalized solutions for a Sobolev critical Schrödinger equation

Math Ann, 2022, 384: 101-134

Jeanjean L, Lu S.

A mass supercritical problem revisited

Calc Var Partial Differential Equations, 2020, 59: Article 174

Jeanjean L, Zhang J, Zhong X.

A global branch approach to normalized solutions for the Schrödinger equation

J Math Pures Appl, 2024, 183: 44-75

Kirchhoff G. Mechanik. Leipzig: Teubner, 1883

Li G, Luo X, Yang T.

Normalized solutions to a class of Kirchhoff equations with Sobolev critical exponent

Ann Fenn Math, 2022, 47: 895-925

Li G, Ye H.

On the concentration phenomenon of $L^2$-subcritical constrained minimizers for a class of Kirchhoff equations with potentials

J Differential Equations, 2019, 266: 7101-7123

Li Q, Rădulescu V, Zhang J, Zhao X.

Normalized solutions of the autonomous Kirchhoff equation with Sobolev critical exponent: sub-and super-critical cases

Proc Amer Math Soc, 2023, 151: 663-678

Lions J.

On some questions in boundary value problems of mathematical physics

North-Holland Math Stud, 1978, 30: 284-346

Lions P.

The concentration-compactness principle in the calculus of variations

The locally compact case, part 1. Ann Inst H Poincaré Anal Non Linéaire, 1984, 1: 109-145

Liu L, Chen H, Yang J.

Normalized solutions to the fractional Kirchhoff equations with a perturbation

Appl Anal, 2023, 102: 1229-1249

Liu Z.

Multiple normalized solutions for Choquard equations involving Kirchhoff type perturbation

Topol Methods Nonlinear Anal, 2019, 54: 297-319

Luo X, Wang Q.

Existence and asymptotic behavior of high energy normalized solutions for the Kirchhoff type equations in $\mathbb{R}^3$

Nonlinear Anal: Real World Appl, 2017, 33: 19-32

Mo S, Ma S.

Normalized solutions to Kirchhoff equation with nonnegative potential

arXiv: 2301.07926

Qi S.

Normalized solutions for the Kirchhoff equation on noncompact metric graphs

Nonlinearity, 2021, 34: 6963-7004

Qi S, Zou W.

Exact number of positive solutions for the Kirchhoff equation

SIAM J Math Anal, 2022, 54: 5424-5446

Soave N.

Normalized ground states for the NLS equation with combined nonlinearities

J Differential Equations, 2020, 269: 6941-6987

Soave N.

Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case

J Funct Anal, 2020, 279: 108610

Wang Q, Qian A.

Normalized solutions to the Kirchhoff equation with potential term: Mass super-critical case

Bull Malays Math Sci Soc, 2023, 46: Article 77

Wang Z.

Existence and asymptotic behavior of normalized solutions for the modified Kirchhoff equations in $\mathbb{R}^3$

J AIMS Math, 2022, 7: 8774-8801

Wei J, Wu Y.

Normalized solutions for Schrödinger equations with critical Sobolev exponent and mixed nonlinearities

J Funct Anal, 2022, 283: 109574

Weinstein M.

Nonlinear Schrödinger equations and sharp interpolation estimates

Comm Math Phys, 1983, 87: 567-576

Xie W, Chen H.

Existence and multiplicity of normalized solutions for the nonlinear Kirchhoff type problems

Comput Math Appl, 2018, 76: 579-591

Yang Z.

Normalized ground state solutions for Kirchhoff type systems

J Math Phys, 2021, 62: 031504

Ye H.

The sharp existence of constrained minimizers for a class of nonlinear Kirchhoff equations

Math Methods Appl Sci, 2015, 38: 2663-2679

Ye H.

The existence of normalized solutions for $L^2$-critical constrained problems related to Kirchhoff equations

Z Angew Math Phys, 2015, 66: 1483-1497

Ye H.

The mass concentration phenomenon for $L^2$-critical constrained problems related to Kirchhoff equations

Z Angew Math Phys, 2016, 67: Article 29

Zeng X, Zhang J, Zhang Y, Zhong X.

On the Kirchhoff equation with prescribed mass and general nonlinearities

Discrete Contin Dyn Syst Ser S, 2023, 16: 3394-3409

Zeng X, Zhang Y.

Existence and uniqueness of normalized solutions for the Kirchhoff equation

Appl Math Lett, 2017, 74: 52-59

Zhang J, Zhang J, Zhong X.

Normalized solutions to Kirchhoff type equations with a critical growth nonlinearity

arXiv: 2210.12911

Zhang P, Han Z.

Normalized ground states for Kirchhoff equations in $\mathbb{R}^3$ with a critical nonlinearity

J Math Phys, 2022, 63: 021505

Zhu X, Li F, Liang Z.

Normalized solutions of a transmission problem of Kirchhoff type

Calc Var Partial Differential Equations, 2021, 60: Article 192

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